= Solution
Under the <open set condition> there is a nonempty bounded <open set> $O$ such that the sets $S_i(O)$ are pairwise disjoint and all contained in $O$. The <Hausdorff dimension> of the nonempty <compact> <invariant set> is the unique $s\geq0$ satisfying
$$
\boxed{\sum_{i=1}^k c_i^s=1.}
$$
For the carpet here the 24 retained sub-squares give <Euclidean similarities> $S_{ij}(x,y)=((x+i)/5,(y+j)/5)$ for $0\leq i,j\leq4$, $(i,j)\ne(2,2)$. Their images of $O=(0,1)^2$ are disjoint and lie in $O$, so the condition holds even though some closed images share boundary points. All contraction factors are $1/5$. Thus
$$
\boxed{\dim_H X=\frac{\log24}{\log5}\approx1.97464.}
$$
Back to article page
